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<h1>Change waist $w_0$</h1>

<h2>Lock waist</h2>

Calculate the beam radius $w$ and the wavefront ROC $R$ at given distance $z$.

$$ z_0 = \frac{\pi w_0^2}{M^2 \lambda}  $$

$$ V_s = \frac{M^2 \lambda}{\pi w_0}  $$

$$ w = w_0 \sqrt{ 1 + \bigg( \frac{z}{z_0} \bigg)^2 } $$

$$ R = z \Bigg[ 1 + \bigg( \frac{z_0}{z} \bigg)^2 \Bigg] $$

$$ q^{-1} = \frac 1 R + i \frac{\lambda}{\pi w^2} $$

<h2>Lock front</h2>

For specified waist radius $w_0$, find such a distance $z$ and beam quality $M^2$ at which the wavefront ROC keeps its previously calculated value.

How to find: express $z_0$ from $w(z)$ and from $R(z)$, equate both expressions and solve against $z$.

$$ z = R \Bigg[ 1 - \bigg( \frac{w_0}{w} \bigg)^2 \Bigg] $$

$$ z_0 = \sqrt{ \frac{ z^2 w_0^2 }{ w^2 - w_0^2 } } $$

$$ z_0 = \sqrt{ z (R - z) }  $$

$$ M^2 = \frac{ \pi w_0^2 }{ \lambda z_0 }  $$

$$ V_s = \frac{M^2 \lambda}{\pi w_0}  $$


<h1>Change Rayleigh length $z_0$</h1>

<h2>Lock waist</h2>

With fixed waist radius $w_0$, find a value of the beam quality parameter $M^2$ which yields the specified $z_0$.

$$ M^2 = \frac{ \pi w_0^2 }{ \lambda z_0 }  $$

$$ V_s = \frac{M^2 \lambda}{\pi w_0}  $$

$$ w = w_0 \sqrt{ 1 + \bigg( \frac{z}{z_0} \bigg)^2 } $$

$$ R = z \Bigg[ 1 + \bigg( \frac{z_0}{z} \bigg)^2 \Bigg] $$

$$ q^{-1} = \frac 1 R + i \frac{\lambda}{\pi w^2} $$

<h2>Lock front</h2>

For specified $z_0$, find such a distance $z$ and beam quality $M^2$ at which the wavefront ROC keeps its previously calculated value. $+$ sign is for the far zone, $-$ is for the near zone.

$$ z = \frac R 2 \pm \frac{\sqrt{ R^2 - 4 z_0^2 }}2  $$

$$ w_0 = \sqrt{ \frac{w^2}{1 + \big( z/z_0 \big)^2} } $$

$$ M^2 = \frac{ \pi w_0^2 }{ \lambda z_0 }  $$

$$ V_s = \frac{M^2 \lambda}{\pi w_0}  $$


<h1>Change divergence angle $V_s$</h1>

<h2>Lock waist</h2>

With fixed waist radius $w_0$, find such a value of beam quality parameter $M^2$ that yields to the specified angle.

$$ M^2 = \frac{ \pi w_0 V_s } \lambda $$

$$ z_0 = \frac{\pi w_0^2}{M^2 \lambda}  $$

$$ w = w_0 \sqrt{ 1 + \bigg( \frac{z}{z_0} \bigg)^2 } $$

$$ R = z \Bigg[ 1 + \bigg( \frac{z_0}{z} \bigg)^2 \Bigg] $$

$$ q^{-1} = \frac 1 R + i \frac{\lambda}{\pi w^2} $$

<h2>Lock front</h2>

For specified $V_s$, find such a distance $z$ and beam quality $M^2$ at which the wavefront ROC keeps its previously calculated value.

How to find $z$: substitute $z_0$ as $w_0/V_s$ into formulas for $w(z)$ and $R(z)$, express $w_0$ from both, equate them and solve against $z$.

$$ z = \frac{w^2}{R V_s^2} $$

$$ z_0 = \sqrt{z (R - z)}  $$

$$ w_0 = \frac{w}{\sqrt{ 1 + \big( z/z_0 \big)^2} } $$

$$ M^2 = \frac{\pi w_0^2}{z_0 \lambda} $$


<h1>Change axial distance $z$</h1>

<h2>Lock waist</h2>

$$ z_0 = \frac{\pi w_0^2}{M^2 \lambda}  $$

$$ V_s = \frac{M^2 \lambda}{\pi w_0}  $$

$$ w = w_0 \sqrt{ 1 + \bigg( \frac{z}{z_0} \bigg)^2 } $$

$$ R = z \Bigg[ 1 + \bigg( \frac{z_0}{z} \bigg)^2 \Bigg] $$

$$ q^{-1} = \frac 1 R + i \frac{\lambda}{\pi w^2} $$

<h2>Lock front</h2>

$$ z_0 = \sqrt{z (R - z)}  $$

$$ w_0 = \frac{w}{\sqrt{ 1 + \big( z/z_0 \big)^2} } $$

$$ M^2 = \frac{\pi w_0^2}{z_0 \lambda} $$

$$ V_s = \frac{M^2 \lambda}{\pi w_0}  $$


<h1>Change beam quality $M^2$</h1>

<h2>Lock waist</h2>

The same as when the waist is changed.

$$ z_0 = \frac{\pi w_0^2}{M^2 \lambda}  $$

$$ V_s = \frac{M^2 \lambda}{\pi w_0}  $$

$$ w = w_0 \sqrt{ 1 + \bigg( \frac{z}{z_0} \bigg)^2 } $$

$$ R = z \Bigg[ 1 + \bigg( \frac{z_0}{z} \bigg)^2 \Bigg] $$

$$ q^{-1} = \frac 1 R + i \frac{\lambda}{\pi w^2} $$

<h2>Lock front</h2>

Taking a new value of $M^2$, find such a distance $z$ and waist radius $w_0$ preserving constant wavefront.

How to find $z$: express $w_0^2$ from formula for $R(z)$, substitute the expression into formula for $w(z)$ and solve it against $z$.

$$ z = \frac{w^4 \pi^2 R}{\big(M^2 \lambda R\big)^2 + w^4 \pi^2} $$

$$ z_0 = \sqrt{z (R - z)}  $$

$$ w_0 = \frac{w}{\sqrt{ 1 + \big( z/z_0 \big)^2} } $$

$$ V_s = \frac{M^2 \lambda}{\pi w_0}  $$


<h1>Change beam radius $w$</h1>

<h2>Lock waist</h2>

Find beam quality parameter $M^2$ giving specified beam radius at the same axial distance $z$ and with the same waist radius $w_0$.

$$ z_0 = \frac{z w_0}{\sqrt{w^2 - w_0^2}} $$

$$ M^2 = \frac{\pi w_0^2}{z_0 \lambda} $$

$$ V_s = \frac{M^2 \lambda}{\pi w_0}  $$

$$ R = z \Bigg[ 1 + \bigg( \frac{z_0}{z} \bigg)^2 \Bigg] $$

$$ q^{-1} = \frac 1 R + i \frac{\lambda}{\pi w^2} $$

<h2>Lock front</h2>

Find a waist radius $w_0$ and beam quality parameter $M^2$ giving specified beam radius at the same axial distance $z$.

$$ z_0 = \sqrt{z (R - z)}  $$

$$ w_0 = \frac{w}{\sqrt{ 1 + \big( z/z_0 \big)^2} } $$

$$ M^2 = \frac{\pi w_0^2}{z_0 \lambda} $$

$$ V_s = \frac{M^2 \lambda}{\pi w_0}  $$

$$ q^{-1} = \frac 1 R + i \frac{\lambda}{\pi w^2} $$


<h1>Change wavefront ROC $R$</h1>

<h2>Lock waist</h2>

Find beam quality parameter $M^2$ giving specified ROC at the same axial distance $z$ and with the same waist radius $w_0$.

$$ z_0 = \sqrt{z (R - z)}  $$

$$ M^2 = \frac{\pi w_0^2}{z_0 \lambda} $$

$$ V_s = \frac{M^2 \lambda}{\pi w_0}  $$

$$ w = w_0 \sqrt{ 1 + \bigg( \frac{z}{z_0} \bigg)^2 } $$

$$ q^{-1} = \frac 1 R + i \frac{\lambda}{\pi w^2} $$

<h2>Lock front</h2>

Find a waist radius $w_0$ and beam quality parameter $M^2$ giving specified ROC at the same axial distance $z$.

$$ z_0 = \sqrt{z (R - z)}  $$

$$ w_0 = \frac{w}{\sqrt{ 1 + \big( z/z_0 \big)^2} } $$

$$ M^2 = \frac{\pi w_0^2}{z_0 \lambda} $$

$$ V_s = \frac{M^2 \lambda}{\pi w_0}  $$

$$ q^{-1} = \frac 1 R + i \frac{\lambda}{\pi w^2} $$


<h1>Change complex beam parameter $q$</h1>

Assigning either real or imaginary part of complex beam parameter, we define both the beam radius $w$ and the wavefront ROC $R$ at the same time.

$$ R = \frac{1}{Re\big(q^{-1}\big)} $$

$$ w = \sqrt{\frac{\lambda}{\pi Im\big(q^{-1}\big)}} $$

<h2>Lock waist</h2>

Having fixed waist radius $w_0$, find axial distance $z$ and beam quality $M^2$ at which specified front is achieved.

$$ z = R \Bigg[ 1 - \bigg( \frac{w_0}{w} \bigg)^2 \Bigg] $$

$$ z_0 = \sqrt{z (R - z)}  $$

$$ M^2 = \frac{\pi w_0^2}{z_0 \lambda} $$

$$ V_s = \frac{M^2 \lambda}{\pi w_0}  $$

<h2>Lock front</h2>

$$ z_0 = \sqrt{z (R - z)}  $$

$$ w_0 = \frac{w}{\sqrt{ 1 + \big( z/z_0 \big)^2} } $$

$$ M^2 = \frac{\pi w_0^2}{z_0 \lambda} $$

$$ V_s = \frac{M^2 \lambda}{\pi w_0}  $$

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